# Cartesian Product is Small

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## Theorem

Let $a$ and $b$ be small classes.

Then their Cartesian product $\left({a \times b}\right)$ is small:

$\mathscr M \left({a \times b}\right)$

## Proof

 $\ds \left({a \times b}\right)$ $=$ $\ds \left\{ {\left({x, y}\right) : x \in a \land y \in b}\right\}$ Definition of Cartesian Product $\ds$ $=$ $\ds \left\{ {\left({x, y}\right) : \left\{ {x}\right\} \subseteq a \land \left\{ {y}\right\} \subseteq b} \right\}$ Singleton of Element is Subset $\ds$ $\subseteq$ $\ds \left\{ {\left({x, y}\right) : \left\{ {x, y}\right\} \subseteq \left({a \cup b}\right)}\right\}$ Set Union Preserves Subsets $\ds$ $\subseteq$ $\ds \left\{ {\left({x, y}\right) : \left({x, y}\right) \subseteq \mathcal P \left({a \cup b}\right)}\right\}$ Power Set of Subset and Subset Relation is Transitive

So by the definition of power set:

$\left({a \times b}\right) \subseteq \mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$

By Union of Small Classes is Small, $\left({a \cup b}\right)$ is small.

By the Axiom of Powers, $\mathcal P \left({\mathcal P \left({a \cup b}\right)}\right)$ is small.

By Axiom of Subsets Equivalents, $\left({a \times b}\right)$ is small.

$\blacksquare$